Those that downsize just after interest rates rise in order to spend the value locked in their house might lose a bit depending on whether any of the money is reinvested in something else after they sell.
People who will lose the most are house builders. Their revenue is directly tied to house prices. However, for a rate rise to be sustainable there has to be other good ways to make money out there otherwise rates will likely quickly go down again.
My intuition is that it would also precipitate an economic downturn. Many households will see a drastic deterioration of their balance sheets, which should act as a damper on consumer spending. Which would, in theory, have 2nd order effects on the real estate market...
In the late 1970s you could get 17% return on your savings account and mortgage rates were well above 10%. People still bought houses.